Incorporating Decision Maker Preference in Multi-Objective Evolutionary Algorithms Lily Rachmawati B Eng Hons., NUS A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF ELECTRICAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2008 Acknowledgements There are many people whom I wish to thank for the support they have rendered me throughout the course of the Doctoral program I gratefully acknowledge the financial support National University of Singapore during the course of the program I sincerely thank my supervisor, Dr Dipti Srinivasan, for the suggestions and encouragements that helped shape the research direction I also thank the many anonymous referees, whose valuable feedback have contributed greatly to the research work accomplished I am also thankful to the family and friends that have supported me in direct and indirect means from the beginning of the course I would like to extend my especial gratitude to my parents, sister and brother, for the countless advice and encouragement they have given Many thanks as well to Ms Jessica Kusuma, Ms Riyanti Teresa, Mr Lisman Komaladi, Mr Arief Adhitya, Mr Steven Halim, and many others who have offered prayers, encouragement and support during the writing and revision of this thesis Finally, and most importantly, I would like to thank the almighty God for His enduring grace and love i Contents Introduction 1.1 Motivation of Research 1.2 Objectives and Scope of Research 1.3 Contribution of the Research 1.4 Outline of the Thesis 11 Concepts and Terminology 12 2.1 Multi-objective Optimization Problem 12 2.2 Pareto Optimality 13 2.3 Pareto Dominance 14 2.4 Decision Maker Preference 16 2.5 Conclusion 17 Decision Making Preference in Multi Objective Optimization 18 3.1 Features of Human Decision Making 20 3.2 Desirable Properties of Preference-based Evolutionary Search 23 3.3 Approaches based on the Relative Importance of Objectives 28 3.3.1 MAUT-based Algorithms 29 3.3.2 Lexicographic Ordering 37 3.3.3 Outranking-based Algorithms 38 3.4 Approaches based on a Goal/Reference Vector 41 3.5 Approaches based on Optimality of Trade Off 45 ii 3.6 50 3.6.1 Population-based Indicator 50 3.6.2 3.7 Other Approaches Fuzzy Logic 50 51 3.7.1 a priori and Interactive Preference Incorporation 52 3.7.2 3.8 Issues in Preference Incorporation into Evolutionary Algorithms Coevolutionary and Fitness-based Preference Integration 56 Conclusion 58 Fitness Functions in Multi objective Evolutionary Algorithms 60 4.1 Deriving Total Order from Partial Order 62 4.2 Aggregative Fitness Functions 66 4.2.1 Random and Dynamic Weighted Aggregation 66 4.2.2 Maximin Function 67 Dominance-based ranking with niching 68 4.3.1 NSGA-II 70 4.3.2 SPEA2 71 4.3.3 PAES 71 4.3.4 NPGA 72 4.3.5 Distributed Pairwise Comparison 73 4.3.6 SEAMO 74 4.3.7 ε-MOEAs 74 4.3.8 Steady State Replacement 76 4.3 4.4 Conclusion Imprecise Goal Vectors 78 79 5.1 Fuzzy Sets 80 5.2 Preference Representation 81 5.2.1 Goal Vectors 82 5.2.2 Degree of Imprecision 83 5.2.3 Invalid Imprecise Goal Vectors 83 Preference Elicitation 84 5.3 iii 5.4 Preference Integration 85 5.5 Empirical Evaluation 89 5.6 Conclusion 95 Knee Solutions 97 6.1 Preference Representation 99 6.2 Weighted Sum Niching 100 6.3 Parallel Local Weighted-Sum Based Optimization 103 6.3.1 Stage-One Optimization: Information Gathering 104 6.3.2 Estimation of potential knee solutions 105 6.3.3 Stage-Two Optimization: Integration of Preference into MOEA 6.3.4 108 Interactive Preference Articulation 110 6.4 Computational complexity 111 6.5 Empirical Evaluation 113 6.5.1 Performance Metric 114 6.5.2 Simulation Details 118 6.5.3 Covergence, Accuracy and Diversity: A Comparative Analysis 124 6.5.4 Weighted-sum Niching and Parallel Local Weighted-Sum Optimization: Further Remarks 130 6.6 Conclusion 134 Relative Importance of Objectives 7.1 137 Preference Representation 139 7.1.1 Preference structure 139 7.1.2 Mathematical Interpretation 141 7.2 Elicitation of Preference 146 7.3 Integration of Preference information into MOEA 149 7.4 Empirical Evaluation 151 7.4.1 Simulation Details 152 iv 7.4.2 7.5 Result and Discussion 157 Conclusion 161 Conclusion 8.1 163 Contributions 163 8.1.1 8.1.2 Optimality of Trade-Off 164 8.1.3 8.2 Imprecise Goal Vector 164 Relative Importance of Objectives 165 Future Work 166 A Conditions of MAUT 168 B Preference Functions 169 C Steady State MOEA Results 171 D Elicitation Algorithm 180 v Abstract The objective of the majority of work in Evolutionary Multi Objective Optimization (EMOO) is to develop Multi-Objective Evolutionary Algorithms (MOEA) able to find a well-distributed approximation of the Pareto optimal front An evenly distributed set of non dominated solutions equips the decision maker with the trade-off behavior associated with the problem at hand so that (s)he could select from the set the most suitable solutions The aposteriori manual application of human preference to the selection of solutions is rendered impractical in higher dimensional problems as a very large number of solutions would be required to approximate the entire extent of the Pareto front meaningfully An apriori/interactive incorporation of preference information into a MOEA averts the problem by concentrating on a subset of the Pareto front The approach avails the decision maker with a higher resolution in the region of interest in the objective space and aids the progression of the elite population towards the true optima Human preference in multi-objective decision making contains uncertainties and anomalies that are to be taken into account in a formal model of preference The uniqueness of the evolutionary computation approach renders the direct adoption of modelling and implementation techniques developed for classical optimization approaches unsuitable This thesis documents research effort into the articulation and incorporation of preference information into EMOO Models of preference formulated in terms of the importance ranking of objectives, an imprecisely specified reference vector, and objective trade off and their implementations in MOEAs are reviewed Three preference incorporation schemes encompassing the representation, elicitation and implementation of preference information are also proposed The approaches are designed for easy adoption into major state-of-the-art general-purpose MOEAs The first guides the population of solutions towards an imprecisely specified goal vector The second directs the search to regions of optimum trade-off in the Pareto front The third incorporates importance ranking of objective functions into MOEAs The proposed mathematical model of preference caters to incomparability and features a functional correspondence between explicated importance ranking of objectives and a specific subset of the Pareto front The preference elicitation algorithm devised facilitates scalable explication of preference The preference incorporation techniques are validated in an empirical study that involves difficult test problems and comparison with similar preference-based algorithms as well as baseline MOEAs Preferencebased performance metrics are also proposed where applicable to measure the concord between obtained solutions and explicated preference ii List of Figures 2.1 Ideal and Nadir objective vectors 14 3.1 Selection of best solution 25 3.2 Guided dominance (Shaded: area dominated by Xi ) 31 3.3 Dotted: Region dominated by solution, Stripes: Region nondominated with respect to solution 3.4 33 Dotted: Region dominated by solution, Stripes: Region nondominated with respect to solution 34 3.5 Knee Solution 45 3.6 Knee solution: a bulge of the Pareto front 47 3.7 Knee in Concave Region 49 4.1 Block Diagram: Evolutionary algorithm 61 4.2 Pareto dominance imposes partial order on the objective space 64 4.3 Bi-objective example with Maximin Fitness (given in brackets) 68 5.1 Case 1: Vector T lying in dominating region 87 5.2 Case : Vector T lying in non-dominated region 87 5.3 Case : Vector T lying in dominated region 88 5.4 Pareto front of problem ZDT1 91 5.5 Pareto front of problem ZDT2 91 5.6 Pareto front of problem ZDT3 92 5.7 Results for problem ZDT1 94 i 5.8 Results for problem ZDT2 95 5.9 Results for problem ZDT3 95 5.10 Results for problem ZDT4 96 6.1 Knee Region 101 6.2 Multiple Knee Regions 102 6.3 Block Diagram 104 6.4 Test Problem DEB2DK-2 with K = to 121 6.5 Captures of the population at different instants in stage (DO2DK, K=2, s=1) 131 6.6 Result obtained for test problem DO2DK (K=1,s=0) with parallel local optimization (δ = 0.1, 0.2, 0.5) 132 6.7 Result obtained for test problem DO2DK (K=2,s=1) with proposed approach (δ = 0.1, 0.2, 0.5) 133 6.8 Result obtained for test problem DO2DK (K=4,s=1) with proposed approach (δ = 0.1, 0.2, 0.5) 133 6.9 Result obtained for test problem DO2DK (K=2,s=1) with weighted sum niching in [92] 134 6.10 Result obtained for test problem DO2DK (K=4,s=1) with weighted sum niching in [92] 134 7.1 Left: f1 P f2 , Middle: f1 If2 , Right: f2 P f1 142 7.2 List and graphs of c1 and c2 143 7.3 Example 1: Three dimensional view (target subset highlighted) 144 7.4 Example 1: View from f1 -f2 plane (target subset highlighted) 144 7.5 Example 1: View from f1 -f3 plane (target subset highlighted) 145 7.6 Example 1: View from f2 -f3 plane (target subset highlighted) 145 7.7 List and graphs of c1 , c2 and c3 146 7.8 Pareto front of problem DTLZ2 154 7.9 Pareto front of problem DTLZ5 157 7.10 Generational Distance in Six Objective Problems 160 ii prefmat[i][index2] = prefmat[i][index1]; if (relation==1)/*case: index1 P index2, insert below index1*/ /*if maximum then just insert*/ maxi = -1; j=1; for (j=1;j¡(size+1);j++) if (prefmat[i][index1]¡prefmat[i][j]) maxi = -100; if (prefmat[i][j]==(prefmat[i][index1]+1)) maxi = j; if (maxi==-1) prefmat[i][index2]=prefmat[i][index1]+1; else /* not so easy case */ dummy = infix(prefmat,size, i, maxi,index2, 1); if ((all).chains2[i] == 1) /*insert obj index1 into the chains*/ if (relation==2)/*case: indifference*/ prefmat[i][index1] = prefmat[i][index2]; if (relation==1)/*case: preferred*/ if(prefmat[i][index2]==1) /*easy case, insert at head of chain*/ for(j=1;j¡(size+1);j++) if (prefmat[i][j]¿0) prefmat[i][j]=prefmat[i][j]+1; prefmat[i][index1]=1; else /* infix insert, not so easy*/ j=1; while(prefmat[i][j]!=(prefmat[i][index2]-1) AND j¡(size+1)) j+=1; dummy = infix(prefmat, size, i,j,index1,2); return 1; void removeduplicate(int** prefmat, int size) int i,j,k, count[size]; for(i=0;i¡size;i++) if (prefmat[i][0]¿0) for(k=0;k¡size;k++) if (prefmat[k][0]¿0 AND k!=i) j=1; 183 while ((prefmat[i][j]-prefmat[k][j]) ¿=0 AND (j¡size+1)) j+=1; if (j==(size+1)) for (j=0;j¡(size+1);j++) prefmat[k][j]=0; int main () int obj1,obj2,i,j,nobj, quit, rel, counter; char binrel; int **S; FILE *fptinputs; FILE *fptcomp; quit = ’n’; fptinputs = fopen(”userinputs.out”,”w”); fptcomp = fopen(”completepref.out”,”w”); fprintf(fptinputs,”# This file contains the user-input preference”); fprintf(fptcomp,”# This file contains the complete preference information”); printf(” No of objective functions : ”); scanf (” while (nobj ¡2) printf(” You entered printf(” No of objective functions : ”); scanf (” /* initialize matrix S*/ S = (int**) malloc(nobj*sizeof(int*)); for (i=0;i¡nobj;i++) S[i] = malloc((nobj+1)*sizeof(int)); for (j=0;j¡(nobj+1);j++) if ((i+1)==j —— j==0) S[i][j]=1; else S[i][j] = 0; printf(” Start entering user preference information in the format: m R n ”); printf(” m, n : integers indices of objective functions defined from to printf(” R : the binary relation between the two objectives m and n ”); printf(” Preference ”); printf(” Indifference ”); quit = 2; counter = 1; while (quit!=1) /*input binary relations */ 184 printf(” Enter index of the first objectives: ”); scanf(” while (obj1¡1 —— obj1¿nobj) printf(” Value of objective index should be between and printf(” Enter index of the first objectives: ”); scanf(” printf(” Enter index of the second objectives: ”); scanf(” while (obj2¡1 —— obj2¿nobj) printf(” Value of objective index should be between and printf(” Enter index of the second objectives: ”); scanf(” printf(” Enter the binary relation: ”); scanf(” /*analyze if the input is valid, insert into pref matrix*/ i = insert(S,nobj, obj1, obj2, rel); /* remove all substrings */ removeduplicate(S, nobj); /* interpret the relation */ if (rel==1) binrel = ’P’; else if (rel==2) binrel = ’I’; /* write into the input list*/ fprintf(fptinputs,”input # /*print out the partial ordering list */ for (i=0;i¡nobj;i++) if (S[i][0]==1)/*active chain, print out content*/ printf(””); for (j=1;j¡(nobj+1);j++) printf(” /*further inputs*/ printf(” Quit (1 yes /2 no) ?”); scanf(” counter += 1; 185 Bibliography [1] Arrow, J., ”Social Choice and Individual Values”, Cowles Commission Monograph, vol 12, Wiley, New York, 1951 [2] Balling, R., ”The Maximin Fitness Function; Multiobjective City and Regional Planning,” in C.M Fonseca et al 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Decision Making Preference in Multi Objective Optimization 18 3.1 Features of Human Decision Making 20 3.2 Desirable Properties of Preference- based Evolutionary Search... Importance of Objectives The relative importance of objectives is an operationally intuitive notion of decision making preference in multi objective optimization The importance of objectives